The Bargmann-Wigner Equation

The Bargmann-Wigner Equation:
Filed equation for arbitrary spin
Farzad Qassemi
IPM School and Workshop on Cosmology
2-9 June, IPM, Tehran, Iran

Motivation
◦ De-Broglie fusion theory of light:
1) It was motivated by De Broglie to derive
Maxwell’s equation as emergence field
equation of two spin ½ massless
particles (Neutrino in that time)
2) Bargmann and Wigner in their seminal
paper in 1948 group theoretically
speaking, generalize this approach to
the field equation for arbitrary spin

Review
◦ Dirac equation:
(γ.
∂ + m)ψ =
0
◦ Set of Gamma matrices
I
, γ , γ , γ γ , σ
5 µ 5 µ µν
Note: they form a complete set.

Charge conjugation operator
*
Cψ =ψ
C
−1
γ
5 5
Cγ =
T
C γ C
−1
= −γ
µ µ

Procedure
1) We symmetries spinor of rank “n” ,
2) It satisfies Dirac equation for each
spinor index:
ψ αβ..
( γ.
∂ + m)
ψ αα α ' β ...
'
=
0

Spin-1 Particle
◦ Group theoretically speaking, the
symmetric state of two spin-½
particles gives spin-1particle. B-W
method helps us to derive the field
equation corresponding to new field
equation. In spin-1 case it means
we can derive Maxwell’s (Proca)
equation from Dirac equation!

Spin-1
ψ
µ
µν
αβ
= A ( γ σ
µ
C)
αβ
+ F (
µνC)
αβ
◦ Now we contract above equation with;
( C
γ ) ,( C γ ) ,( C )
−1 −1
−1
5 βδ 5γ ν βδ
βδ

Spin-3/2 (Rarita-Schwinger eq)
◦ We can continue to make a spin 3/2
particle as follow;
ραβ
µ
ρ
µ
αβ
µν
ψ = A ( γ C)
+ F ( σ C)
ρ
µν
αβ
◦ Follow the same method of contraction.

Modified BW
◦ Q: We examine if fully antisymmetrization
spinor gives us
what we expect.
◦ A: Yes.

Spin-0
◦ Here we try to (re)derive Klein-
Gordon equation from second rank
anti-symmetric spinor.
◦ The general method is as before,
except we changed the expansion of
the spinor with respect to antisymmetric
gamma matrices.

Spin-0
ψ = A + )
αβ
ν
( γ
5C)
αβ
+ B ( γ
5γνC)
αβ
F(
C
αβ
◦ Now, we contract above spinor with
symmetric gamma matrices.
( C
−1 −1
µ
)
βδ
,( C σ
µν
γ )
βδ

Summary
◦ Starting from Dirac equation one
can derive higher spin field
◦ By modifying BW we examine
validity of the Anti-Symmetric state

Thank You
Thank You